3.5.72 \(\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \, dx\) [472]

Optimal. Leaf size=100 \[ \frac {1}{2} b \left (3 a^2+2 b^2\right ) x+\frac {a \left (2 a^2+9 b^2\right ) \sin (c+d x)}{3 d}+\frac {7 a^2 b \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^2(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{3 d} \]

[Out]

1/2*b*(3*a^2+2*b^2)*x+1/3*a*(2*a^2+9*b^2)*sin(d*x+c)/d+7/6*a^2*b*cos(d*x+c)*sin(d*x+c)/d+1/3*a^2*cos(d*x+c)^2*
(a+b*sec(d*x+c))*sin(d*x+c)/d

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Rubi [A]
time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3926, 4132, 2717, 4130, 8} \begin {gather*} \frac {a \left (2 a^2+9 b^2\right ) \sin (c+d x)}{3 d}+\frac {1}{2} b x \left (3 a^2+2 b^2\right )+\frac {7 a^2 b \sin (c+d x) \cos (c+d x)}{6 d}+\frac {a^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]

[Out]

(b*(3*a^2 + 2*b^2)*x)/2 + (a*(2*a^2 + 9*b^2)*Sin[c + d*x])/(3*d) + (7*a^2*b*Cos[c + d*x]*Sin[c + d*x])/(6*d) +
(a^2*Cos[c + d*x]^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3926

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^2*Co
t[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*n)), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x]
)^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2*(n + 1))*Csc[e + f*x] - b*(b^2*
n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2]
&& ((IntegerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))

Rule 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rubi steps

\begin {align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \, dx &=\frac {a^2 \cos ^2(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) \left (7 a^2 b+a \left (2 a^2+9 b^2\right ) \sec (c+d x)+b \left (a^2+3 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \cos ^2(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) \left (7 a^2 b+b \left (a^2+3 b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} \left (a \left (2 a^2+9 b^2\right )\right ) \int \cos (c+d x) \, dx\\ &=\frac {a \left (2 a^2+9 b^2\right ) \sin (c+d x)}{3 d}+\frac {7 a^2 b \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^2(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac {1}{2} \left (b \left (3 a^2+2 b^2\right )\right ) \int 1 \, dx\\ &=\frac {1}{2} b \left (3 a^2+2 b^2\right ) x+\frac {a \left (2 a^2+9 b^2\right ) \sin (c+d x)}{3 d}+\frac {7 a^2 b \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^2(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{3 d}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 80, normalized size = 0.80 \begin {gather*} \frac {18 a^2 b c+12 b^3 c+18 a^2 b d x+12 b^3 d x+9 a \left (a^2+4 b^2\right ) \sin (c+d x)+9 a^2 b \sin (2 (c+d x))+a^3 \sin (3 (c+d x))}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]

[Out]

(18*a^2*b*c + 12*b^3*c + 18*a^2*b*d*x + 12*b^3*d*x + 9*a*(a^2 + 4*b^2)*Sin[c + d*x] + 9*a^2*b*Sin[2*(c + d*x)]
+ a^3*Sin[3*(c + d*x)])/(12*d)

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Maple [A]
time = 0.10, size = 76, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 b \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 b^{2} a \sin \left (d x +c \right )+b^{3} \left (d x +c \right )}{d}\)	\(76\)
default	\(\frac {\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 b \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 b^{2} a \sin \left (d x +c \right )+b^{3} \left (d x +c \right )}{d}\)	\(76\)
risch	\(\frac {3 a^{2} b x}{2}+b^{3} x +\frac {3 a^{3} \sin \left (d x +c \right )}{4 d}+\frac {3 \sin \left (d x +c \right ) b^{2} a}{d}+\frac {a^{3} \sin \left (3 d x +3 c \right )}{12 d}+\frac {3 b \,a^{2} \sin \left (2 d x +2 c \right )}{4 d}\)	\(78\)
norman	\(\frac {\left (\frac {3}{2} b \,a^{2}+b^{3}\right ) x +\left (\frac {3}{2} b \,a^{2}+b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} b \,a^{2}+b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} b \,a^{2}+b^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 b \,a^{2}-2 b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 b \,a^{2}-2 b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a \left (2 a^{2}-3 b a +6 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (2 a^{2}+3 b a +6 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a \left (a^{2}-9 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{2} \left (4 a -9 b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{2} \left (4 a +9 b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}\)	\(302\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3*a^3*(2+cos(d*x+c)^2)*sin(d*x+c)+3*b*a^2*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+3*b^2*a*sin(d*x+c)+
b^3*(d*x+c))

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Maxima [A]
time = 0.27, size = 73, normalized size = 0.73 \begin {gather*} -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b - 12 \, {\left (d x + c\right )} b^{3} - 36 \, a b^{2} \sin \left (d x + c\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/12*(4*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^3 - 9*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^2*b - 12*(d*x + c)*b^3 -
36*a*b^2*sin(d*x + c))/d

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Fricas [A]
time = 2.87, size = 66, normalized size = 0.66 \begin {gather*} \frac {3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} d x + {\left (2 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{2} b \cos \left (d x + c\right ) + 4 \, a^{3} + 18 \, a b^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/6*(3*(3*a^2*b + 2*b^3)*d*x + (2*a^3*cos(d*x + c)^2 + 9*a^2*b*cos(d*x + c) + 4*a^3 + 18*a*b^2)*sin(d*x + c))/
d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \cos ^{3}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(a+b*sec(d*x+c))**3,x)

[Out]

Integral((a + b*sec(c + d*x))**3*cos(c + d*x)**3, x)

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Giac [A]
time = 0.47, size = 170, normalized size = 1.70 \begin {gather*} \frac {3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/6*(3*(3*a^2*b + 2*b^3)*(d*x + c) + 2*(6*a^3*tan(1/2*d*x + 1/2*c)^5 - 9*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 18*a*b
^2*tan(1/2*d*x + 1/2*c)^5 + 4*a^3*tan(1/2*d*x + 1/2*c)^3 + 36*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*a^3*tan(1/2*d*x
+ 1/2*c) + 9*a^2*b*tan(1/2*d*x + 1/2*c) + 18*a*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d

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Mupad [B]
time = 0.86, size = 77, normalized size = 0.77 \begin {gather*} b^3\,x+\frac {3\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {3\,a^2\,b\,x}{2}+\frac {3\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,a\,b^2\,\sin \left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^3*(a + b/cos(c + d*x))^3,x)

[Out]

b^3*x + (3*a^3*sin(c + d*x))/(4*d) + (a^3*sin(3*c + 3*d*x))/(12*d) + (3*a^2*b*x)/2 + (3*a^2*b*sin(2*c + 2*d*x)
)/(4*d) + (3*a*b^2*sin(c + d*x))/d

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